A fluid of constant density, $,$ and viscosity, $,$ is contained in a very long horizontal pipe of length $L$ and radius $R.$ Initially, the fluid is at rest. At $t=0,$ a pressure gradient $\frac{p_0-p_L}{L}$ is impressed on the system. Can the method of Laplace transform be used to determine how the velocity profiles change with time?

Hint: Use cylindrical coordinates and assume that the $z-$component of velocity, $v_z,$ is a function of $r$ and $t,$ while the $r-$ and $-$components of velocity are both zero. Then combine the equations of continuity and motion to get $[1]$

$\frac{del{v}_z}{\text{d}\text{e}\text{l}\text{t}}=\frac{p_0-p_L}{L}+\frac{1}{r}\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{r}}(r\frac{del{v}_z}{\text{d}\text{e}\text{l}\text{r}})$

subject to the conditions:

$v_z(r,0)=0$

$v_z(0,t)is$ finite

$v_z(R,t)=0$